I don't see why you can't just take an Access client installer from Server2, put it into the Toolbox of Server1, and then push it out to every client on Server1.

Problem found
How do we keep the name and organisation of the computer using a generic access installer?
I would need to create an installer for each customer's computer with their computer name and organisation it seems since the screenconnect client name is different from the windows machine name.

You're looking at a few hours labor, but preserving it all can definitely be done.

You can write a setup stub with Inno Setup that will parse the registry for the target computers. In the Inno Setup you'll read and parse the contents of the current SC service information, which would be the ImagePath REG_EXPAND_SZ from something like "HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\ScreenConnect Client (1234567890)". Since it's all coming from one SC server, this registry path will be consistent across the entire collection of target computers, but the ImagePath value will not be. Parse the ImagePath to obtain the GUID (s) from the existing installations. Use this as a command-line assignment on the install to any of the property values (which we'll use later to identify it to ensure that updated data goes to the right device).

Then export your database values from the source SC server to Excel. (Excel would be good since you can automate the process of triggering changes on the server.) A little VBA in Excel to push the Property's and Notes to the new SC server and an extension to receive and update them with the new values.